Skip to main content

Full text of "Scientific Papers - Vi"

See other formats


4                       NOTE ON BESSEL'S FUNCTIONS AS APPLIED
since by (3)
   n) +   n  n = - n In (5) all the denominators are positive.    We deduce
 2 _ 2                      ,? 2 _ OT2         -. 2 _ 2
gl           ^   __  1     ,   *1 _ 11    I    ^           ^      ,               ^ I    
*
and therefore
z? >n2 + '2n>n(n + 1).
Our theorems are therefore proved.
If a closer approximation to z-f is desired, it may be obtaine stituting on the right of (6) 2n for z?  n2 in the numerators and i n2 in the denominators. Thus
r\          -'   J-   T     *""
2n
^   + 2 ,,!    , 2     *s    i ...    ,/,~L~O\ r 
ft ^l + 4; J
Now, as is easily proved from the ascending series for Jn',
71+2
ty  ~"2   _J_   /y   "*2      1_       "~~2   _l                                                      
so that finally
w3
z? >n-+ 2- +,-
When n is very great, it will follow from (7) that z? > v/,2 + the approximation is not close, for the ultimate form is*
^ = 7is+[1-6130] n4*
As has been mentioned, the sequence formula In
Jn (g) = Jn_1 (z] + J
n+l
prohibits the simultaneous evanescence of J"n_1 and Jn, or of Jn-i The question arisescan Bessel's functions whose orders (supposei differ by more than 2 vanish simultaneously ?    If we change n in (8) and then eliminate Jn, we get
_
- " 1 ~1          '"
tl+2>
from which it appears that if Jn^ and Jn+2 vanish simultaneously, t Jn+i = 0, which is impossible, or ^-' = 4<n (n + 1). Any common rt and Jn+i must therefore be such that its square is an integer.
* Phil. Mag. Vol. xx. p. 1003, 1910, equation (8).    [1913.   A correction is hen See Nicholson, Phil. Mag. Vol. xxv. p. 200, 1913.]emains to prove that z* necessarily exceeds n(n 4-1). That zli exceeds n2 is well known*, but this does not suffice. We can obtain what we require from a formula given, in Theory of Sound, 2nd ed.  339. If the finite roots taken in order be zi} zz,... za..., we may write